Problem: Determine how many solutions exist for the system of equations. ${-15x-3y = 15}$ ${y = -1-5x}$
Solution: Convert both equations to slope-intercept form: ${-15x-3y = 15}$ $-15x{+15x} - 3y = 15{+15x}$ $-3y = 15+15x$ $y = -5-5x$ ${y = -5x-5}$ ${y = -1-5x}$ ${y = -5x-1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x-5}$ ${y = -5x-1}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.